Square Roots....fun facts

[Mizen]

If you know some interesting or fun facts about square roots, I would love to know them.

 

I have few so far:

 

- square root of 2 is the diagonal of a square who's side length = 1; square root of 3 is the diagonal of a cube who's side length =1

- a list of the most popular square roots and their answers

 

any information that will inspire others to learn more about square roots is appreciated!

Edited March 14, 2012 by Mizen
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[Joatmon]

The square roots of all numbers which are not perfect squares are irrational. There are lots of them! http://en.wikipedia.org/wiki/Irrational_number

[Mizen]

Thanks Joatmon

 

some others:

 

-square root of any area yields side length of a square with same area

-similarly, cube root of any volume yields side length of cube with same volume

 

 

[Joatmon]

If I draw a straight line and ask you to draw a line which represents its square root you cannot do it. If however I give you two lines and ask you to draw a line which represents the square root of their sums squared it's easy. (Right angled triangle).

The apparently easy problem is impossible and the more complicated problem is easy!

Edited March 14, 2012 by Joatmon

[michel123456]

If I draw a straight line and ask you to draw a line which represents its square root you cannot do it. If however I give you two lines and ask you to draw a line which represents the square root of their sums squared it's easy. (Right angled triangle).

The apparently easy problem is impossible and the more complicated problem is easy!

Bolded mine

 

It is the square root of the sum of their squares. [math]A^2+B^2=C^2[/math]

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The square root of natural numbers can be constructed geometricaly

 

Edited March 14, 2012 by michel123456

[Joatmon]

Bolded mine

 

It is the square root of the sum of their squares. [math]A^2+B^2=C^2[/math]

Yes, of course you are right. I knew what I meant but it came out wrong! The main fact I wanted to get over was the more complicated problem was easily solved whereas the apparently more simple problem was ,in fact, impossible.

Thanks for putting things right.

 

 

[phillip1882]

err, the easily worded problem...

If I draw a straight line and ask you to draw a line which represents its square root[...]

is actually quite possible.

asumming you have a unit length line, and a given line, the square root will simply be the concatination of the two lines, the mid point of the distance between them used to make a circle, and a perpendicular line from the meeting point to the circle.

[imatfaal]

In these sorts of constructions I do not think you can have access to a unit line. Straight edge, compass and paper

 

unless you know how to extend a line by 1/19 (in your example) by only using the edge, and compass

[phillip1882]

here's a fun one. the golden ratio; 1.618... is equal to...

sqrt(1+sqrt(1+sqrt(1+...

or more simply (1+sqrt(5))/2

newton provided an interesting method for calculating the square root; namely

 

x1 = x0 -(x0 ^2 -a)/(2*x0)

 

which increases the accuracy of the aproximation quadratically.

as an example, sqrt(27) =?

initial guess, 5.

x1 = 5 - (25-27)/10

x1 = 5.2

x2 = 5.2 -(27.04 -27)/10.4

x2 = 5.196

 

err i think you have to have access to a unit line; other wise who's to say the 19 isn't 1?

also even if i didnt, i can divide any line an equal number of times, so i certainly can contruct a line of length 1 given a line of length 19.

 

 

 

Edited April 20, 2012 by phillip1882

[John Cuthber]

The square roots of all numbers which are not perfect squares are irrational. There are lots of them! http://en.wikipedia.org/wiki/Irrational_number

Except for the ones that are not.

Root (9/16) = 3/4 which is rational.

However, I think you are right for integers.

[zapatos]

The square root of 69 is 8 something.

[Joatmon]

The square root of 69 is 8 something.

So is 69 itself (or so I have been told!). No need to complicate things.

 

Except for the ones that are not.

Root (9/16) = 3/4 which is rational.

However, I think you are right for integers.

 

Loose writing - sorry lol. Of course I was thinking integers.